{"id":"n-c1f6a6363c73beb4","revision":1,"first_index":0,"fingerprint":"f6-cbfa13cb29880cff","name":"reversal of Triangle, row sums = number of edges in n-dimensional hypercubes.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["3","1"],["5","6","1"],["7","15","9","1"],["9","28","30","12","1"],["11","45","70","50","15","1"],["13","66","135","140","75","18","1"],["15","91","231","315","245","105","21","1"],["17","120","364","616","630","392","140","24","1"],["19","153","540","1092","1386","1134","588","180","27","1"],["21","190","765","1800","2730","2772","1890","840","225","30","1"],["23","231","1045","2805","4950","6006","5082","2970","1155","275","33","1"],["25","276","1386","4180","8415","11880","12012","8712","4455","1540","330","36","1"],["27","325","1794","6006","13585","21879","25740","22308","14157","6435","2002","390","39","1"],["29","378","2275","8372","21021","38038","51051","51480","39039","22022","9009","2548","455","42","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","12","32","80","192","448","1024","2304","5120","11264","24576","53248","114688","245760"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","-3"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163538,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-c51898909065bf38"]},{"id":163539,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-c51898909065bf38"]},{"id":163540,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-c51898909065bf38"]}],"region_exact":{},"incoming_edges":[{"id":"e-c51898909065bf38","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-c1f6a6363c73beb4","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"a947013124bed2515dfb70a20ea39d439d36f9d142521b5aef4a63917fb66e04","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6202e291a2627bce","role":"sequence","name":"Triangle, row sums = number of edges in n-dimensional hypercubes.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-2c964cddd0638543","kind":"candidate_equal","node_a_id":"n-698a3f7f3e3b2c70","node_b_id":"n-c1f6a6363c73beb4","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-698a3f7f3e3b2c70","other":{"id":"n-698a3f7f3e3b2c70","name":"shift_x_plus_one of Triangle read by rows, (n-1) zeros followed by (2n, 1).","first_index":0,"fingerprint":"f6-cbfa13cb29880cff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["5","6","1"]],"card_rows":[["1"],["3","1"],["5","6","1"],["7","15","9","1"]],"row_sums_preview":["1","4","12","32","80","192","448","1024"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["2D+1","x \u21a6 x+1"],"edges":["e-224384a428b81acf","e-fdc4f030d71c09e5"],"text":"A023531 \u2192 2D+1 \u2192 x \u21a6 x+1"},"headline":"A023531 \u2192 2D+1 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-7c783464c2756767","kind":"fingerprint_related","node_a_id":"n-6202e291a2627bce","node_b_id":"n-c1f6a6363c73beb4","fingerprint":"f6-cbfa13cb29880cff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cbfa13cb29880cff"},"other_id":"n-6202e291a2627bce","other":{"id":"n-6202e291a2627bce","name":"Triangle, row sums = number of edges in n-dimensional hypercubes.","first_index":0,"fingerprint":"f6-cbfa13cb29880cff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","3"],["1","6","5"]],"card_rows":[["1"],["1","3"],["1","6","5"],["1","9","15","7"]],"row_sums_preview":["1","4","12","32","80","192","448","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A116666"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["2\u03b8+1"],"edges":["e-c4ea45619bc7f60a"],"text":"A007318 \u2192 2\u03b8+1"},"headline":"A116666: Triangle, row sums = number of edges in n-dimensional hypercubes.","chain":"A007318 \u2192 2\u03b8+1"}},{"id":"l-07b49436bc2cc294","kind":"oeis_match","node_a_id":"n-c1f6a6363c73beb4","node_b_id":null,"fingerprint":null,"oeis_id":"A134083","in_catalog":0,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":false,"oeis_id":"A134083","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-15071282ba52a52f","kind":"oeis_match","node_a_id":"n-c1f6a6363c73beb4","node_b_id":null,"fingerprint":null,"oeis_id":"A097067","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A097067","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-694fbb7756d1e098","kind":"oeis_match","node_a_id":"n-c1f6a6363c73beb4","node_b_id":null,"fingerprint":null,"oeis_id":"A001787","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001787","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bb44a91dab2509cf","kind":"oeis_match","node_a_id":"n-c1f6a6363c73beb4","node_b_id":null,"fingerprint":null,"oeis_id":"A116666","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":61},"in_catalog":true,"oeis_id":"A116666","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c523ceab4467991b","kind":"oeis_match","node_a_id":"n-c1f6a6363c73beb4","node_b_id":null,"fingerprint":null,"oeis_id":"A139756","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A139756","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A116666","relation":"match_reversed_triangle"},{"oeis_id":"A001787","relation":"match_row_sums"},{"oeis_id":"A097067","relation":"match_row_sums"},{"oeis_id":"A139756","relation":"match_row_sums"},{"oeis_id":"A134083","relation":"match_triangle"}],"bfiles":[{"node_id":"n-c1f6a6363c73beb4","oeis_id":"A116666","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":61,"oeis_terms_agreeing":61,"oeis_terms_available":61,"term_count":990,"raw_size":11922},{"node_id":"n-c1f6a6363c73beb4","oeis_id":"A134083","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":11920}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  3,   1;\n  2:  5,   6,   1;\n  3:  7,  15,   9,    1;\n  4:  9,  28,  30,   12,    1;\n  5: 11,  45,  70,   50,   15,    1;\n  6: 13,  66, 135,  140,   75,   18,   1;\n  7: 15,  91, 231,  315,  245,  105,  21,   1;\n  8: 17, 120, 364,  616,  630,  392, 140,  24,  1;\n  9: 19, 153, 540, 1092, 1386, 1134, 588, 180, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(4*n-3)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (4*n - 3)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 1 + (4*n - 3)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(2*n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A116666","target":"reversed_triangle","first_row":0,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  3;\n   3: 1,  6,   5;\n   4: 1,  9,  15,   7;\n   5: 1, 12,  30,  28,    9;\n   6: 1, 15,  50,  70,   45,   11;\n   7: 1, 18,  75, 140,  135,   66,   13;\n   8: 1, 21, 105, 245,  315,  231,   91,  15;\n   9: 1, 24, 140, 392,  630,  616,  364, 120,  17;\n  10: 1, 27, 180, 588, 1134, 1386, 1092, 540, 153, 19;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n-3)*P(n,x) = (2*n-1+(4*n-7)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 1 + (4*n - 7)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 1 + (4*n - 7)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(2*n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A134083","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:  1;\n  1:  3,   1;\n  2:  5,   6,   1;\n  3:  7,  15,   9,    1;\n  4:  9,  28,  30,   12,    1;\n  5: 11,  45,  70,   50,   15,    1;\n  6: 13,  66, 135,  140,   75,   18,   1;\n  7: 15,  91, 231,  315,  245,  105,  21,   1;\n  8: 17, 120, 364,  616,  630,  392, 140,  24,  1;\n  9: 19, 153, 540, 1092, 1386, 1134, 588, 180, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = (2*n+1+(4*n-3)*x)*P(n-1,x) - (2*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 1 + (4*n - 3)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x])/(2*n - 1)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 1 + (4*n - 3)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x))/(2*n - 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-c8c19eba9d68881f","steps":["2\u03b8+1","reverse"],"edges":["e-c4ea45619bc7f60a","e-c51898909065bf38"],"text":"A007318 \u2192 2\u03b8+1 \u2192 reverse"},"headline":"A007318 \u2192 2\u03b8+1 \u2192 reverse","chain":null}
